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Pre-Calculus · course sample

Pre-Calculus Essentials

Connect function families, logarithms, trigonometry, and sequences while paying attention to domains and mathematical reasoning.

Grades 10–12 · accounts 13+5 chapters · 15 lesson goalsRead-only · no account needed
About this read-only sample

Authored illustrative samples. These static, read-only outlines and examples were created for this website, not extracted from production courses. They have not been teacher-reviewed and are not a verified Florida curriculum.

Grade ranges describe the material, not account eligibility. Younger-grade samples are public and read-only; accounts require ages 13+. Children under 13 should not sign up or submit personal information.

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The full sample outline

Start with the foundations, then build on one idea at a time.

  1. Chapter 1

    Functions

    Build a toolkit for comparing and combining mathematical rules.

    1. 1

      Domain and transformations

      Track how shifts, stretches, and restrictions change a function's graph.

    2. 2

      Composition

      Evaluate a function of another function in the correct order.

    3. 3

      Inverse functions

      Explain when an inverse is a function and how to check it.

  2. Chapter 2

    Polynomial and rational functions

    Connect algebraic structure to zeros, restrictions, and graph behavior.

    1. 1

      Polynomial zeros and end behavior

      Relate factors, degree, and leading coefficient to a graph.

    2. 2

      Rational expressions and domains

      Identify excluded inputs before simplifying an expression.

    3. 3

      Asymptotes and holes

      Distinguish a removable discontinuity from a vertical asymptote in simple examples.

  3. Chapter 3

    Exponentials and logarithms

    Use inverse relationships to describe multiplicative change.

    1. 1

      Exponential models

      Interpret a growth or decay factor and state the model's assumptions.

    2. 2

      Logarithms as exponents

      Translate between logarithmic and exponential statements.

    3. 3

      Solve and verify

      Use logarithm rules appropriately and reject inputs outside the domain.

  4. Chapter 4

    Trigonometry

    Connect angles, coordinates, and periodic patterns.

    1. 1

      Radians and the unit circle

      Relate common angles to coordinates on the unit circle.

    2. 2

      Trigonometric graphs

      Interpret amplitude, period, and horizontal shifts.

    3. 3

      Identities and equations

      Use basic identities and report all solutions in a specified interval.

  5. Chapter 5

    Sequences

    Describe a numerical pattern with a rule instead of only guessing its next term.

    1. 1

      Arithmetic sequences

      Find a common difference and write an explicit rule.

    2. 2

      Geometric sequences

      Find a common ratio and distinguish multiplicative from additive change.

    3. 3

      Recursive and explicit rules

      Compare two ways of specifying a sequence and identify the starting index.

Let's work through an example

Order matters in composition

Let f(x) = 2x + 1 and g(x) = x². Find f(g(3)).

  1. 1

    Start with the inside function: g(3) = 3² = 9. The output of g becomes the input to f.

  2. 2

    Substitute that output into f: f(9) = 2 × 9 + 1 = 19. Therefore f(g(3)) = 19.

  3. 3

    Check the order: g(f(3)) would instead be g(7) = 49. Reversing the functions generally changes the result.

The idea to keep: Composition is not multiplication. Read from the innermost input outward, and check that each input is allowed.

Check your thinking · read-only quiz

For f(x) = 2x + 1 and g(x) = x², what is f(g(2))?

Choose an answer in your head, then open the explanation. Nothing is submitted, graded, or saved.

  1. A.5
  2. B.9
  3. C.16
  4. D.25
Reveal answer and explanation

Answer B: 9

First evaluate g(2) = 2² = 4. Then evaluate f(4) = 2 × 4 + 1 = 9. The result 25 would come from reversing the order and finding g(f(2)).